A virtually stable US unemployment rate for December has not eased organized labor's worries about high and prolonged joblessness in 1980. Union leaders are paying more attention to spreading layoffs in auto, steel, and other manufacturing industries -- and a 13 percent jump in claims for state unemployment benefits -- as warning signs of a worsening situation.

Under the circumstances, says Lane Kirkland, president of the AFL-CIO, President Carter's recent order to tighten rules for unemployment compensation for the long- term jobless is untimely and "could actually further deepen the recession with unemployment feeding on unemployment."

President Carter on Jan. 5 ordered changes that are expected to reduce the number of jobless workers who receive federally assisted benefits. The White House says the move is designed to save the government hundreds of millions of dollars needed in 1980 and 1981 to stimulate housing.

Organized labor has softened its criticism of President Carter in recent months largely because of the crises in Iran and Afghanistan. The White House order could have political repercussions in the presidential primaries later this year. Unions blame Mr. Carter's economic policies for the rising unemployment rate: A substantial reduction in the ranks of those receiving aid will add new fuel to the fires of protest.

The AFL-CIO vehemently opposes changes in the formula used for making extended benefits available to those who exhaust state jobless benefits. Labor Secretary F. Ray Marshall also argued against any change at this time, as did the National Governors Association. Employers and most state employment security administrators favored the change, effective Feb. 3, however, and the President ordered it made.

Administration supporters say the new regulations are based on forecasts of only a mild recession and an unemployment rate between 7 and 7.5 percent by this fall.

Outside predictions run higher, to 8 percent or more, even though unemployment figures through December were surprisingly strong and stable. While the national jobless rate nudged upward from 5.8 to 5.9 percent in December, the number holding jobs rose by 304,000 to bring the total employed to 97,912,000.

The latest increase raised the number of working Americans to 59.4 percent of the population. At 5.9 percent, the number unemployed and seeking jobs was 6, 087,000.

Janet L. Norwood, commissioner of Labor Statistics, said that the unemployment rate has shown "remarkable stability" for 17 months. However, she warned that the December jobless figure did not reflect substantial layoffs occurring in auto and other industries.